Pre-Algebra Examples

$9 x^{2} + 5 y^{2} - 18 x + 30 y + 9 = 0$

Step 1

Find the standard form of the ellipse.

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Step 1.1

Subtract $9$ from both sides of the equation.

$9 x^{2} + 5 y^{2} - 18 x + 30 y = - 9$

Step 1.2

Complete the square for $9 x^{2} - 18 x$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 9$

$b = - 18$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 18}{2 \cdot 9}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 18$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 18$ .

$d = \frac{2 \cdot - 9}{2 \cdot 9}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 9}{2 (9)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 9}{2 \cdot 9}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 9}{9}$

Step 1.2.3.2.2

Cancel the common factor of $- 9$ and $9$ .

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Step 1.2.3.2.2.1

Factor $9$ out of $- 9$ .

$d = \frac{9 \cdot - 1}{9}$

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 1}{9 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 1}{9 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 1}{1}$

Step 1.2.3.2.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 18)}^{2}}{4 \cdot 9}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raise $- 18$ to the power of $2$ .

$e = 0 - \frac{324}{4 \cdot 9}$

Step 1.2.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{324}{36}$

Step 1.2.4.2.1.3

Divide $324$ by $36$ .

$e = 0 - 1 \cdot 9$

Step 1.2.4.2.1.4

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Step 1.2.4.2.2

Subtract $9$ from $0$ .

$e = - 9$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 1)}^{2} - 9$ .

$9 {(x - 1)}^{2} - 9$

Step 1.3

Substitute $9 {(x - 1)}^{2} - 9$ for $9 x^{2} - 18 x$ in the equation $9 x^{2} + 5 y^{2} - 18 x + 30 y = - 9$ .

$9 {(x - 1)}^{2} - 9 + 5 y^{2} + 30 y = - 9$

Step 1.4

Move $- 9$ to the right side of the equation by adding $9$ to both sides.

$9 {(x - 1)}^{2} + 5 y^{2} + 30 y = - 9 + 9$

Step 1.5

Complete the square for $5 y^{2} + 30 y$ .

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Step 1.5.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 5$

$b = 30$

$c = 0$

Step 1.5.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.5.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{30}{2 \cdot 5}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $30$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $30$ .

$d = \frac{2 \cdot 15}{2 \cdot 5}$

Step 1.5.3.2.1.2

Cancel the common factors.

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Step 1.5.3.2.1.2.1

Factor $2$ out of $2 \cdot 5$ .

$d = \frac{2 \cdot 15}{2 (5)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 15}{2 \cdot 5}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{15}{5}$

Step 1.5.3.2.2

Cancel the common factor of $15$ and $5$ .

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Step 1.5.3.2.2.1

Factor $5$ out of $15$ .

$d = \frac{5 \cdot 3}{5}$

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $5$ out of $5$ .

$d = \frac{5 \cdot 3}{5 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{5 \cdot 3}{5 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

$d = \frac{3}{1}$

Step 1.5.3.2.2.2.4

Divide $3$ by $1$ .

$d = 3$

Step 1.5.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{30^{2}}{4 \cdot 5}$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Simplify each term.

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Step 1.5.4.2.1.1

Raise $30$ to the power of $2$ .

$e = 0 - \frac{900}{4 \cdot 5}$

Step 1.5.4.2.1.2

Multiply $4$ by $5$ .

$e = 0 - \frac{900}{20}$

Step 1.5.4.2.1.3

Divide $900$ by $20$ .

$e = 0 - 1 \cdot 45$

Step 1.5.4.2.1.4

Multiply $- 1$ by $45$ .

$e = 0 - 45$

Step 1.5.4.2.2

Subtract $45$ from $0$ .

$e = - 45$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(y + 3)}^{2} - 45$ .

$5 {(y + 3)}^{2} - 45$

Step 1.6

Substitute $5 {(y + 3)}^{2} - 45$ for $5 y^{2} + 30 y$ in the equation $9 x^{2} + 5 y^{2} - 18 x + 30 y = - 9$ .

$9 {(x - 1)}^{2} + 5 {(y + 3)}^{2} - 45 = - 9 + 9$

Step 1.7

Move $- 45$ to the right side of the equation by adding $45$ to both sides.

$9 {(x - 1)}^{2} + 5 {(y + 3)}^{2} = - 9 + 9 + 45$

Step 1.8

Simplify $- 9 + 9 + 45$ .

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Step 1.8.1

Add $- 9$ and $9$ .

$9 {(x - 1)}^{2} + 5 {(y + 3)}^{2} = 0 + 45$

Step 1.8.2

Add $0$ and $45$ .

$9 {(x - 1)}^{2} + 5 {(y + 3)}^{2} = 45$

Step 1.9

Divide each term by $45$ to make the right side equal to one.

$\frac{9 {(x - 1)}^{2}}{45} + \frac{5 {(y + 3)}^{2}}{45} = \frac{45}{45}$

Step 1.10

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{{(x - 1)}^{2}}{5} + \frac{{(y + 3)}^{2}}{9} = 1$

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

$\frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1$

Step 3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = 3$

$b = \sqrt{5}$

$k = - 3$

$h = 1$

Step 4

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(1, - 3)$

Step 5

Find $c$ , the distance from the center to a focus.

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Step 5.1

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Step 5.2

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(3)}^{2} - {(\sqrt{5})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Raise $3$ to the power of $2$ .

$\sqrt{9 - {(\sqrt{5})}^{2}}$

Step 5.3.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{9 - {(5^{\frac{1}{2}})}^{2}}$

Step 5.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{9 - 5^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{9 - 5^{\frac{2}{2}}}$

Step 5.3.2.4

Cancel the common factor of $2$ .

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Step 5.3.2.4.1

Cancel the common factor.

$\sqrt{9 - 5^{\frac{2}{2}}}$

Step 5.3.2.4.2

Rewrite the expression.

$\sqrt{9 - 5^{1}}$

Step 5.3.2.5

Evaluate the exponent.

$\sqrt{9 - 1 \cdot 5}$

Step 5.3.3

Simplify the expression.

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Step 5.3.3.1

Multiply $- 1$ by $5$ .

$\sqrt{9 - 5}$

Step 5.3.3.2

Subtract $5$ from $9$ .

$\sqrt{4}$

Step 5.3.3.3

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2}}$

Step 5.3.4

Pull terms out from under the radical, assuming positive real numbers.

$2$

Step 6

Find the vertices.

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Step 6.1

The first vertex of an ellipse can be found by adding $a$ to $k$ .

$(h, k + a)$

Step 6.2

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(1, - 3 + 3)$

Step 6.3

Simplify.

$(1, 0)$

Step 6.4

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Step 6.5

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(1, - 3 - (3))$

Step 6.6

Simplify.

$(1, - 6)$

Step 6.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(1, 0)$

${Vertex}_{2}$ : $(1, - 6)$

${Vertex}_{1}$ : $(1, 0)$

${Vertex}_{2}$ : $(1, - 6)$

Step 7

Find the foci.

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Step 7.1

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Step 7.2

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(1, - 3 + 2)$

Step 7.3

Simplify.

$(1, - 1)$

Step 7.4

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 7.5

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(1, - 3 - (2))$

Step 7.6

Simplify.

$(1, - 5)$

Step 7.7

Ellipses have two foci.

${Focus}_{1}$ : $(1, - 1)$

${Focus}_{2}$ : $(1, - 5)$

${Focus}_{1}$ : $(1, - 1)$

${Focus}_{2}$ : $(1, - 5)$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} - b^{2}}}{a}$

Step 8.2

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(3)}^{2} - {(\sqrt{5})}^{2}}}{3}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise $3$ to the power of $2$ .

$\frac{\sqrt{9 - {\sqrt{5}}^{2}}}{3}$

Step 8.3.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{9 - {(5^{\frac{1}{2}})}^{2}}}{3}$

Step 8.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{9 - 5^{\frac{1}{2} \cdot 2}}}{3}$

Step 8.3.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{9 - 5^{\frac{2}{2}}}}{3}$

Step 8.3.2.4

Cancel the common factor of $2$ .

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Step 8.3.2.4.1

Cancel the common factor.

$\frac{\sqrt{9 - 5^{\frac{2}{2}}}}{3}$

Step 8.3.2.4.2

Rewrite the expression.

$\frac{\sqrt{9 - 5^{1}}}{3}$

Step 8.3.2.5

Evaluate the exponent.

$\frac{\sqrt{9 - 1 \cdot 5}}{3}$

Step 8.3.3

Multiply $- 1$ by $5$ .

$\frac{\sqrt{9 - 5}}{3}$

Step 8.3.4

Subtract $5$ from $9$ .

$\frac{\sqrt{4}}{3}$

Step 8.3.5

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}}}{3}$

Step 8.3.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2}{3}$

Step 9

These values represent the important values for graphing and analyzing an ellipse.

Center: $(1, - 3)$

${Vertex}_{1}$ : $(1, 0)$

${Vertex}_{2}$ : $(1, - 6)$

${Focus}_{1}$ : $(1, - 1)$

${Focus}_{2}$ : $(1, - 5)$

Eccentricity: $\frac{2}{3}$

Step 10